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Circles \(X\) and \(Y\) are concentric. If the radius of circle \(X\) is three times that of circle \(Y\), what is the probability that a point selected inside circle \(X\) at random will be outside circle \(Y\)?

A. \(\frac{1}{3}\) B. \(\frac{\pi}{3}\) C. \(\frac{\pi}{2}\) D. \(\frac{5}{6}\) E. \(\frac{8}{9}\)

Circles \(X\) and \(Y\) are concentric. If the radius of circle \(X\) is three times that of circle \(Y\), what is the probability that a point selected inside circle \(X\) at random will be outside circle \(Y\)?

A. \(\frac{1}{3}\) B. \(\frac{\pi}{3}\) C. \(\frac{\pi}{2}\) D. \(\frac{5}{6}\) E. \(\frac{8}{9}\)

We have to find the ratio of the area of the ring around the small circle to the area of the big circle. If \(y\) is the radius of the smaller circle, then the area of the bigger circle is \(\pi(3y)^2 = 9 \pi y^2\). The area of the ring \(= \pi(3y)^2 - \pi(y)^2 = 8 \pi y^2\). The ratio \(= \frac{8}{9}\).